\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 58, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/58\hfil A curve of stable solutions]
{A global curve of stable, positive
solutions for a $p$-Laplacian problem}
\author[B. P. Rynne\hfil EJDE-2010/58\hfilneg]
{Bryan P. Rynne}
\address{Department of Mathematics and the Maxwell Institute for
Mathematical Sciences, Heriot-Watt University,
Edinburgh EH14 4AS, Scotland}
\email{bryan@ma.hw.ac.uk}
\thanks{Submitted August 13, 2009. Published April 28, 2010.}
\subjclass[2000]{34B15}
\keywords{Ordinary differential equations; $p$-Laplacian; \hfill\break\indent
nonlinear boundary value problems; positive solutions; stability}
\begin{abstract}
We consider the boundary-value problem
\begin{gather*}
- \phi_p (u'(x))' = \lambda f(x,u(x)) , \quad x \in (0,1),\\
u(0) = u(1) = 0,
\end{gather*}
where
$p>1$ ($p \ne 2$), $\phi_p(s) := |s|^{p-1} \mathop{\rm sign} s$, $s \in \mathbb{R}$,
$\lambda \ge 0$,
and the function
$f : [0,1] \times \mathbb{R} \to \mathbb{R}$ is $C^1$ and
satisfies
\begin{gather*}
f(x,\xi) > 0, \quad (x,\xi) \in [0,1] \times \mathbb{R} ,\\
(p-1)f(x,\xi) \ge f_\xi(x,\xi) \xi ,
\quad (x,\xi) \in [0,1] \times (0,\infty) .
\end{gather*}
These assumptions on $f$ imply that the trivial solution
$(\lambda,u)=(0,0)$ is
the only solution
with $\lambda=0$ or $u=0$,
and if $\lambda > 0$ then any solution $u$ is {\em positive},
that is, $u > 0$ on $(0,1)$.
We prove that the set of nontrivial solutions
consists of a $C^1$ curve of positive solutions in
$(0,\lambda_{\rm max}) \times C^0[0,1]$,
with a parametrisation of the form
$\lambda \to (\lambda,u(\lambda))$,
where $u$ is a $C^1$ function defined on $(0,\lambda_{\rm max})$,
and $\lambda_{\rm max}$ is a suitable weighted eigenvalue of the
$p$-Laplacian
($\lambda_{\rm max}$ may be finite or $\infty$),
and $u$ satisfies
$$
\lim_{\lambda\to 0} u(\lambda) = 0,
\quad
\lim_{\lambda \to \lambda_{\rm max}} |u(\lambda)|_0 = \infty .
$$
We also show that for each $\lambda \in (0,\lambda_{\rm max})$
the solution $u(\lambda)$ is globally asymptotically stable,
with respect to positive solutions
(in a suitable sense).
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{cor}[thm]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction} \label{intro.sec}
We consider the boundary-value problem
\begin{gather}
- \phi_p (u'(x))' = \lambda f(x,u(x)) , \quad x \in (0,1),
\label{main.eq}\\
u(0) = u(1) = 0,
\label{bc.eq}
\end{gather}
where
$p>1$ ($p \ne 2$), $\phi_p(s) := |s|^{p-1} \mathop{\rm sign} s$,
$s \in \mathbb{R}$, $\lambda \ge 0$,
and the function
$f : [0,1] \times \mathbb{R} \to \mathbb{R}$ is $C^1$ and
satisfies
\begin{gather} \label{posf.eq}
f(x,\xi) > 0, \quad (x,\xi) \in [0,1] \times \mathbb{R} ,\\ \label{fcond.eq}
(p-1)f(x,\xi) \ge f_\xi(x,\xi) \xi ,
\quad (x,\xi) \in [0,1] \times (0,\infty) .
\end{gather}
The condition \eqref{posf.eq} ensures that the trivial solution
$(\lambda,u)=(0,0)$ is the only solution
with $\lambda=0$ or $u=0$,
and if $\lambda > 0$ then any solution $u$ is {\em positive},
that is, $u > 0$ on $(0,1)$.
In the semilinear case ($p=2$) the problem
\eqref{main.eq}--\eqref{bc.eq}
has been considered in many papers, under various hypotheses on $f$,
see for example
\cite{BIS,CR,DANSTR,HER,K1,K2,K3,LAE,SHI,SS}.
When $f$ is independent of $x$, detailed results for this case
are obtained in \cite{BIS} and \cite{LAE}.
These papers use quadrature to derive explicit formulae for a $C^1$
curve of solutions in $[0,\infty) \times C^0[0,1]$, passing through
$(\lambda,u) = (0,0)$, with a parametrisation of the form
$s \to (\lambda(s),u(s))$,
where the parameter $s = |u(s)|_0$.
The results on the shape of the solution curve are then obtained by
investigating the function $s \to \lambda(s)$.
However, when $f$ depends on $x$, such a formula for the solutions is
not available.
Despite this, curves of solutions, with similar properties to those
in the $x$ independent case, have been constructed in, for example,
\cite[Section~4]{CR}, and \cite{HER,K1,K2,K3,LAE,SHI,SS}
(again under a variety of hypotheses on $f$).
In these papers the strategy is to use the implicit
function theorem to construct a solution curve in
$[0,\infty) \times C^0[0,1]$ by continuation away from the solution
$(\lambda,u) = (0,0)$,
and then investigate the structure of this curve directly.
The case of general $p > 1$ ($p \ne 2$) with $f$ independent of $x$ has
been considered in many recent papers using the quadrature method,
see for example,
\cite{AW1,AW2,BA,CS,KS,WY}.
In this paper we consider the general $p$ case, with $f$ dependent on
$x$, and we use the continuation approach to prove the following
results.
There exists $\lambda_{\rm max} > 0$
($\lambda_{\rm max}$ may be finite or $\infty$)
such that the set of nontrivial solutions of
\eqref{main.eq}--\eqref{bc.eq} consists of a $C^1$ curve of
globally stable, positive solutions in
$(0,\lambda_{\rm max}) \times C^0[0,1]$, with a parametrisation of the
form $\lambda \to (\lambda,u(\lambda))$,
where $u$ is a $C^1$ function defined on the interval
$(0,\lambda_{\rm max})$.
Furthermore,
$$
\lim_{\lambda\to 0} \lambda^{-p^*} u(\lambda) = - \Delta_p^{-1}(f(0)),
\quad
\lim_{\lambda \to \lambda_{\rm max}} |u(\lambda)|_0 = \infty ,
$$
(where $\Delta_p^{-1}$ is the inverse of the $p$-Laplacian operator, and
$p^* := (p-1)^{-1}$),
so the curve meets the point $(0,0)$.
We also characterise the value of $\lambda_{\rm max}$ as a
weighted eigenvalue of the $p$-Laplacian
Under the hypotheses
\eqref{posf.eq}, \eqref{fcond.eq},
similar results have been obtained in the semilinear case $p=2$,
and in the general $p$ case with $f$ independent of $x$.
Other hypotheses on $f$ yield so called `S-shaped' curves of solutions
(the form of the parametrisation described above shows that S-shaped
curves are precluded by \eqref{posf.eq}, \eqref{fcond.eq}),
see for example \cite{BIS,K2,WY}, and the references therein.
The continuation approach relies on the use of the implicit function
theorem.
In order to apply the implicit function theorem to the above problem we
require some recent results on differentiability of the inverse
of the $p$-Laplacian. These results will be described in
Section~\ref{K_diff.sec}, and the solution curve will then be
constructed in Section~\ref{main_res.sec}.
In Section~\ref{stability.sec} it will be shown that the solutions on
this curve are globally asymptotically stable with respect to positive
solutions
(in a sense to be made precise below).
Finally, in Section~\ref{bif_curve.sec}, we briefly consider the
situation when we change the condition \eqref{posf.eq} to
allow $f(\cdot,0) = 0$
(while retaining \eqref{fcond.eq}),
and we show that a similar $C^1$ curve of solutions exists.
\section{Preliminaries} \label{prelim.sec}
For any integer $r \ge 0$, $C^r[0,1]$ will denote the standard
Banach
space of real valued, $r$-times continuously differentiable functions
defined on $[0,1]$, with the norm $|u|_r = \sum_{i=0}^r
|u^{(i)}|_0$,
where $|\cdot|_0$ denotes the usual sup-norm on $C^0[0,1]$
(throughout, all function spaces will be real).
For any $q \ge 1$, $L^q(0,1)$ will denote the standard Banach
space of real valued functions on $[0,1]$ whose $q$th power is
integrable, with norm $\|\cdot\|_q$.
We let $W^{1,q}(0,1)$, with norm $\|\cdot\|_{1,q}$,
denote the usual Sobolev space of absolutely continuous
functions $u$ on $[0,1]$, with derivative $u' \in L^q(0,1)$,
while $W_0^{1,q}(0,1)$ denotes the set of functions in $W^{1,q}(0,1)$
satisfying \eqref{bc.eq}.
If $F : X \to Z$ is a function between Banach spaces $X$ and $Z$,
then $Df(x) : X \to Z$ will denote the
Fr\'echet derivative of $F$ at $x$;
partial Fr\'echet derivatives will be indicated by subscripts,
for example, $D_x G(x,y),\, D_y G(x,y)$ will denote the partial
derivatives of a function $G$ depending on $x$ and $y$.
\subsection{The $p$-Laplacian and its inverse} \label{K_diff.sec}
Letting
$$
\mathcal{D}_p : \{ u \in C^1[0,1] :
\text{$u$ satisfies \eqref{bc.eq} and $\phi_p(u') \in W^{1,1}(0,1)$\} ,}
$$
we define the $p$-Laplacian operator $\Delta_p : \mathcal{D}_p \to L^1(0,1)$ by
$$
\Delta_p(u) = \phi_p(u')' , \quad u \in \mathcal{D}_p .
$$
This operator is {\em $(p-1)$-homogeneous}, that is,
$\Delta_p(t u) = t^{p-1} \Delta_p(u)$,
for any $t \in \mathbb{R}$ and $u \in \mathcal{D}_p$.
The following invertibility result
is well known --- see, for example,
\cite[Theorem~3.1]{BR}, \cite[Theorem~20]{HM}
(these references prove the result for periodic boundary conditions, but
the proof can readily be modified to deal with Dirichlet boundary
conditions).
\begin{theorem} \label{De_inv.thm}
For any $h \in L^1(0,1)$, the problem
\begin{equation} \label{De_h.eq}
\Delta_p(u) = h,
\quad h \in L^1(0,1) ,
\end{equation}
has a unique solution $u = \Delta_p^{-1}(h) \in \mathcal{D}_p$.
The operator
$\Delta_p^{-1} : L^1(0,1) \to C^1[0,1]$
is continuous and $p^*$-homogeneous
$($recall that $p^* := (p-1)^{-1})$.
The operator
$\Delta_p^{-1} : L^1(0,1) \to C^0[0,1]$
is compact.
\end{theorem}
Next we discuss the differentiability of the operator $\Delta_p^{-1}$.
The following result is proved in \cite[Theorem~3.4]{BR}
for the periodic case; the proof in the Dirichlet case is similar
(but simpler).
A similar result is described in Theorem~5 and Corollary~6 of
\cite{GS}, however, the arguments in the proofs in \cite{GS}
seem to be incomplete.
\begin{theorem} \label{diff_K.thm}
For $h \in L^1(0,1)$, let $u = u(h) := \Delta_p^{-1}(h)$.
\begin{itemize}
\item[(A)] Suppose that $p>2$
and $h \in C^0[0,1]$ is such that
$u'(x) = 0 \implies h(x) \ne 0$, for $x \in [0,1]$.
Then there exists a neighbourhood $V$ of $h$ in $C^0[0,1]$
such that:
\begin{itemize}
\item[(a)]
for $h \in V$,
$|u(h)'|^{2-p} \in L^1(0,1);$
\item[(b)]
the mapping $h \to |u(h)'|^{2-p} : V \to L^1(0,\pi_p)$
is continuous;
\item[(c)]
the mapping $\Delta_p^{-1} : V \to W_0^{1,1}(0,1)$ is $C^1$
and
\begin{equation} \label{DK_per_diff_form.eq}
w = D\Delta_p^{-1}(h) \bar h \implies ( |u'|^{p-2} w' )' = p^* \bar h ,
\quad \bar h \in C^0[0,1] .
\end{equation}
\end{itemize}
\item[(B)] Suppose that $1